Bradač–Sudakov–Wigderson conjecture for nearly regular locally dense graphs

Let $H$ and $G$ be graphs, let $t(H,G)$ denote the homomorphism density of $H$ in $G$, and call an $n$-vertex graph $G$ $(\rho,d)$-nearly-regular when all but at most $\rho n$ vertices have degrees in $[(d-\rho)n,(d+\rho)n]$. Also call $G$ $(\rho,d)$-dense when every subset $X$ of at least $\rho n$ vertices spans at least $\frac{d}{2}|X|^2$ edges. Bradač–Sudakov–Wigderson conjecture. For every graph $H$ and all real $d,\varepsilon>0$, there exists a $\rho=\rho(d,H,\varepsilon)>0$ such that

$$t(H,G)\geq (1-\varepsilon)\cdot d^{|E(H)|}$$

holds for every sufficiently large $(\rho,d)$-dense $(\rho,d)$-nearly-regular graph $G$. This is a restricted, almost-regular version of the KNRS conjecture. The source presents it as an open conjecture and notes that the authors prove results for subdivisions related to it.